Bounds for Eigenvalues with the Use of Finite Elements

Abstract

Verified upper and lower bounds for the smallest eigenvalues of eigenvalue problems with self-adjoint partial differential equations are computed. Upper bounds are obtained by the Rayleigh-Ritz method, a suitable Goerisch method provides lower bounds. The trial functions are constructed with the use of finite elements. All computations are carried out with intervaI arithmetic thus the results are protected against rounding errors. Numerical results are given for the L-shaped membrane.